A total preorder over a set X is a preorder <= over X that is total, that is, for all a and b in X it holds that a <= b or b <= a.

The difference between a total preorder and a total order is that a total preorder (since it is only a preorder) is not required to be antisymmetric.

An example is (X, <=) where X = {1, 2} and a <= b for all a and b in X.